Calculator Inputs
Example Data Table
This example uses N = 10,000, I0 = 10, R0 = 0, β = 0.30, γ = 0.10, and a 160 day horizon.
| Day | Susceptible | Infected | Recovered | Notes |
|---|---|---|---|---|
| 0 | 9,990 | 10 | 0 | Initial state. |
| 20 | 9,258.48 | 488.04 | 253.48 | Early growth phase. |
| 40 | 2,875.24 | 2,973.27 | 4,151.49 | Near peak spread. |
| 60 | 871.54 | 998.20 | 8,130.26 | Declining infection load. |
| 80 | 641.90 | 208.41 | 9,149.69 | Late epidemic stage. |
| 160 | 594.54 | 0.30 | 9,405.16 | Near final size. |
Formula Used
The calculator follows the standard SIR system. It treats people as susceptible, infected, or recovered. Population size stays fixed during the run.
dS/dt = -βSI/N
dI/dt = βSI/N - γI
dR/dt = γI
The basic reproduction number is R0 = β / γ. The initial effective reproduction number is Re(0) = R0 × S0 / N.
If you choose contact mode, the calculator first converts inputs with β = contact rate × transmission probability.
The estimated final outbreak size comes from the SIR final size relation. It solves the final susceptible count numerically using:
ln(S∞) + (β/γN)(N - S∞) = ln(S0) + (β/γN)R0
Then final outbreak size is N - S∞ - R0. Peak timing and curve shape come from a numerical RK4 simulation across the selected time horizon.
How to Use This Calculator
- Enter a scenario name so exported files stay clear.
- Choose direct beta input or contact based input.
- Provide population, infected, and recovered starting values.
- Enter the recovery rate gamma.
- Enter beta directly, or fill contact rate and transmission probability.
- Set the simulation days and time step.
- Press the calculate button.
- Review outbreak size, attack rate, peak day, and curve data.
- Export the scenario using the CSV or PDF buttons.
About This SIR Outbreak Size Calculator
Model purpose
This calculator estimates how many people are eventually infected in a closed SIR outbreak. It also shows the epidemic path over time. The approach is useful for teaching, planning, and scenario comparison. It fits many introductory epidemic studies and compartment flow examples.
Why the final size matters
Final outbreak size is different from the peak. Peak infected tells you the highest active burden at one time. Final size tells you how many people were infected across the whole outbreak. Both values matter. One helps with capacity. The other helps with total impact.
Inputs that drive the result
Transmission rate beta controls how quickly infections spread. Recovery rate gamma controls how quickly infected people leave the infectious class. Their ratio creates the basic reproduction number. Initial infected and recovered values also matter. They shape the starting condition and can shift the curve.
Why simulation and equation are both used
The final size equation gives a stable estimate for the eventual outbreak total. The numerical simulation adds timing. It shows when infections rise, when the peak happens, and how susceptible and recovered groups move. Using both methods gives a fuller picture. It is practical and easy to interpret.
Physics style interpretation
This framework also resembles flow and decay systems seen in mathematical physics. Susceptible people move into the infected state. Infected people decay into the recovered state. The transfer rates shape the curve. That makes the SIR system a strong teaching example for rate based modeling.
Best use cases
Use this calculator for quick scenario tests, classroom demonstrations, and sensitivity checks. It works best when population mixing is fairly uniform and the system stays closed. It is not a replacement for detailed public health forecasting. It is a clear first pass tool.
FAQs
1) What does outbreak size mean here?
It means the total number of people infected during the full outbreak. It is estimated as the final recovered total minus any people already recovered at the start.
2) Why is the final size different from peak infected?
Peak infected is the highest active infected count on one day. Final size is cumulative impact across the whole outbreak. They answer different planning questions.
3) What is the role of beta?
Beta controls how fast infections spread from contact between susceptible and infected groups. A higher beta usually increases growth speed, peak size, and final outbreak size.
4) What does gamma represent?
Gamma is the recovery rate. It measures how quickly infected people leave the infectious class. A higher gamma shortens infectious duration and often reduces outbreak growth.
5) Can I enter contact rate instead of beta?
Yes. Choose the contact mode. The calculator multiplies contact rate by transmission probability to produce beta before running the model.
6) Why does the model assume a closed population?
That assumption keeps the classic SIR equations valid. It removes births, deaths, migration, and outside introductions so the flow between compartments stays consistent.
7) What does the time step change?
The time step controls numerical resolution. Smaller values usually create smoother curves and slightly better accuracy, but they also increase the number of calculated rows.
8) Is this enough for real world forecasting?
No. It is a simplified educational tool. Real forecasts often need age structure, network effects, interventions, changing behavior, and uncertainty analysis.